Let $S _ { n }$ denote the sum of the first $n$ terms of the arithmetic sequence $\left\{ a _ { n } \right\}$. If $2 S _ { 1 } = 3 S _ { 2 } + 6$ , then the common difference $d = $ $\_\_\_\_$ .

Let $S _ { n }$ denote the sum of the first $n$ terms of the sequence $\left\{ a _ { n } \right\}$. Given that $\frac { 2 S _ { n } } { n } + n = 2 a _ { n } + 1$.
(1) Prove that $\left\{ a _ { n } \right\}$ is an arithmetic sequence;
(2) If $a _ { 4 }$, $a _ { 7 }$, $a _ { 9 }$ form a geometric sequence, find the minimum value of $S _ { n }$.

Let $S _ { n }$ denote the sum of the first $n$ terms of the sequence $\left\{ a _ { n } \right\}$. Given $\frac { 2 S _ { n } } { n } + n = 2 a _ { n } + 1$ .
(1) Prove that $\left\{ a _ { n } \right\}$ is an arithmetic sequence;
(2) If $a _ { 4 } , a _ { 7 } , a _ { 9 }$ form a geometric sequence, find the minimum value of $S _ { n }$ .

In the sequence $\left\{ a_{n} \right\}$ , let $S_{n}$ be the sum of the first $n$ terms of $\left\{ a_{n} \right\}$ , $S_{5} = 5S_{3} - 4$ , then $S_{4} =$
A. $7$
B. $9$
C. $15$
D. $20$

Given that the arithmetic sequence $\left\{ a _ { n } \right\}$ has common difference $\frac { 2 \pi } { 3 }$, and the set $S = \left\{ \cos a _ { n } \mid n \in \mathbb{N} ^ { * } \right\}$. If $S = \{ a , b \}$, then $a b =$
A. $- 1$
B. $- \frac { 1 } { 2 }$
C. 0
D. $\frac { 1 } { 2 }$

Let $S _ { n }$ denote the sum of the first $n$ terms of an arithmetic sequence $\left\{ a _ { n } \right\}$. If $a _ { 3 } + a _ { 4 } = 7$ and $3 a _ { 2 } + a _ { 5 } = 5$, then $S _ { 10 } =$ $\_\_\_\_$ .

(17 points) Let $m$ be a positive integer. The sequence $a _ { 1 } , a _ { 2 } , \cdots , a _ { 4 m + 2 }$ is an arithmetic sequence with nonzero common difference. If after removing two terms $a _ { i }$ and $a _ { j } ( i < j )$ , the remaining $4 m$ terms can be evenly divided into $m$ groups, and the 4 numbers in each group form an arithmetic sequence, then the sequence $a _ { 1 } , a _ { 2 } , \cdots , a _ { 4 m + 2 }$ is called an $( i , j )$ -divisible sequence.
(1) Write out all pairs $( i , j )$ with $1 \leqslant i < j \leqslant 6$ such that the sequence $a _ { 1 } , a _ { 2 } , \cdots , a _ { 6 }$ is an $( i , j )$ -divisible sequence;
(2) When $m \geqslant 3$ , prove that the sequence $a _ { 1 } , a _ { 2 } , \cdots , a _ { 4 m + 2 }$ is a $( 2,13 )$ -divisible sequence;
(3) From $1,2 , \cdots , 4 m + 2$ , randomly select two numbers $i$ and $j$ ( $i < j$ ) at once. Let $P _ { m }$ denote the probability that the sequence $a _ { 1 } , a _ { 2 } , \cdots , a _ { 4 m + 2 }$ is an $( i , j )$ -divisible sequence. Prove that $P _ { m } > \frac { 1 } { 8 }$ .

Let $S_n$ denote the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$. If $S_3 = 6$, $S_5 = -5$, then $S_6 = $ ( )
A. $-20$
B. $-15$
C. $-10$
D. $-5$

Let the sequence $\{a_n\}$ satisfy $a_1 = 3$, $\frac{a_{n+1}}{n} = \frac{a_n}{n+1} + \frac{1}{n(n+1)}$.
(1) Prove that $\{na_n\}$ is an arithmetic sequence.
(2) Let $f(x) = a_1 x + a_2 x^2 + \cdots + a_m x^m$. Find $f'(-2)$.

(15 points) Let the sequence $\{a_n\}$ satisfy $a_1 = 3$, $\frac{a_{n+1}}{n} = \frac{a_n}{n+1} + \frac{1}{n(n+1)}$.
(1) Prove that $\{na_n\}$ is an arithmetic sequence.
(2) Let $f(x) = a_1 x + a_2 x^2 + \cdots + a_m x^m$. Find $f'(-2)$.

Recall Stirling's formula. Deduce an asymptotic equivalent of $C _ { n }$ as $n$ tends to $+ \infty$.

From the previous question, recover the result of questions 11 and 16.

If the $n$ terms $a _ { 1 } , a _ { 2 } , \ldots , a _ { n }$ are in arithmetic progression with increment $r$, then the difference between the mean of their squares and the square of their mean is
(A) $\frac { r ^ { 2 } \left( ( n - 1 ) ^ { 2 } - 1 \right) } { 12 }$
(B) $\frac { r ^ { 2 } } { 12 }$
(C) $\frac { r ^ { 2 } \left( n ^ { 2 } - 1 \right) } { 12 }$
(D) $\frac { n ^ { 2 } - 1 } { 12 }$

In how many ways can we choose $a _ { 1 } < a _ { 2 } < a _ { 3 } < a _ { 4 }$ from the set $\{ 1,2 , \ldots , 30 \}$ such that $a _ { 1 } , a _ { 2 } , a _ { 3 } , a _ { 4 }$ are in arithmetic progression?
(A) 135
(B) 145
(C) 155
(D) 165

If the sum of first $n$ terms of an A.P. is $cn^{2}$, then the sum of squares of these $n$ terms is
(A) $\frac{n\left(4n^{2}-1\right)c^{2}}{6}$
(B) $\frac{n\left(4n^{2}+1\right)c^{2}}{3}$
(C) $\frac{n\left(4n^{2}-1\right)c^{2}}{3}$
(D) $\frac{n\left(4n^{2}+1\right)c^{2}}{6}$

Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots , a _ { 11 }$ be real numbers satisfying $\mathrm { a } _ { 1 } = 15 , \quad 27 - 2 \mathrm { a } _ { 2 } > 0$ and $\mathrm { a } _ { \mathrm { k } } = 2 \mathrm { a } _ { \mathrm { k } - 1 } - \mathrm { a } _ { \mathrm { k } - 2 }$ for $\mathrm { k } = 3,4 , \ldots , 11$.
If $\frac { a _ { 1 } ^ { 2 } + a _ { 2 } ^ { 2 } + \ldots + a _ { 11 } ^ { 2 } } { 11 } = 90$, then the value of $\frac { a _ { 1 } + a _ { 2 } + \ldots + a _ { 11 } } { 11 }$ is equal to

A pack contains $n$ cards numbered from 1 to $n$. Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on the removed cards is $k$, then $k - 20 =$

The probability that $x_1, x_2, x_3$ are in an arithmetic progression, is
(A) $\frac{9}{105}$
(B) $\frac{10}{105}$
(C) $\frac{11}{105}$
(D) $\frac{7}{105}$

Suppose that all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is 6:11 and the seventh term lies in between 130 and 140, then the common difference of this A.P. is

Let $b _ { i } > 1$ for $i = 1,2 , \ldots , 101$. Suppose $\log _ { e } b _ { 1 } , \log _ { e } b _ { 2 } , \ldots , \log _ { e } b _ { 101 }$ are in Arithmetic Progression (A.P.) with the common difference $\log _ { e } 2$. Suppose $a _ { 1 } , a _ { 2 } , \ldots , a _ { 101 }$ are in A.P. such that $a _ { 1 } = b _ { 1 }$ and $a _ { 51 } = b _ { 51 }$. If $t = b _ { 1 } + b _ { 2 } + \cdots + b _ { 51 }$ and $s = a _ { 1 } + a _ { 2 } + \cdots + a _ { 51 }$, then
(A) $s > t$ and $a _ { 101 } > b _ { 101 }$
(B) $s > t$ and $a _ { 101 } < b _ { 101 }$
(C) $s < t$ and $a _ { 101 } > b _ { 101 }$
(D) $s < t$ and $a _ { 101 } < b _ { 101 }$

Let $X$ be the set consisting of the first 2018 terms of the arithmetic progression $1,6,11 , \ldots$, and $Y$ be the set consisting of the first 2018 terms of the arithmetic progression $9,16,23 , \ldots$. Then, the number of elements in the set $X \cup Y$ is $\_\_\_\_$.

Let $A P ( a ; d )$ denote the set of all the terms of an infinite arithmetic progression with first term $a$ and common difference $d > 0$. If $$A P ( 1 ; 3 ) \cap A P ( 2 ; 5 ) \cap A P ( 3 ; 7 ) = A P ( a ; d )$$ then $a + d$ equals $\_\_\_\_$

Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ be a sequence of positive integers in arithmetic progression with common difference 2. Also, let $b _ { 1 } , b _ { 2 } , b _ { 3 } , \ldots$ be a sequence of positive integers in geometric progression with common ratio 2. If $a _ { 1 } = b _ { 1 } = c$, then the number of all possible values of $c$, for which the equality
$$2 \left( a _ { 1 } + a _ { 2 } + \cdots + a _ { n } \right) = b _ { 1 } + b _ { 2 } + \cdots + b _ { n }$$
holds for some positive integer $n$, is $\_\_\_\_$

Let $l _ { 1 } , l _ { 2 } , \ldots , l _ { 100 }$ be consecutive terms of an arithmetic progression with common difference $d _ { 1 }$, and let $w _ { 1 } , w _ { 2 } , \ldots , w _ { 100 }$ be consecutive terms of another arithmetic progression with common difference $d _ { 2 }$, where $d _ { 1 } d _ { 2 } = 10$. For each $i = 1,2 , \ldots , 100$, let $R _ { i }$ be a rectangle with length $l _ { i }$, width $w _ { i }$ and area $A _ { i }$. If $A _ { 51 } - A _ { 50 } = 1000$, then the value of $A _ { 100 } - A _ { 90 }$ is $\_\_\_\_$.

Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ be an arithmetic progression with $a _ { 1 } = 7$ and common difference 8. Let $T _ { 1 } , T _ { 2 } , T _ { 3 } , \ldots$ be such that $T _ { 1 } = 3$ and $T _ { n + 1 } - T _ { n } = a _ { n }$ for $n \geq 1$. Then, which of the following is/are TRUE ?
(A) $T _ { 20 } = 1604$
(B) $\sum _ { k = 1 } ^ { 20 } T _ { k } = 10510$
(C) $T _ { 30 } = 3454$
(D) $\sum _ { k = 1 } ^ { 30 } T _ { k } = 35610$